, Volume 63, Issue 2, pp 221–247 | Cite as

Twisting Hopf algebras from cocycle deformations

  • Nicolás Andruskiewitsch
  • Agustín García Iglesias


Let H be a Hopf algebra. Any finite-dimensional lifting of \(V\in {}^{H}_{H}\mathcal {YD}\) arising as a cocycle deformation of \(A={\mathfrak {B}}(V)\#H\) defines a twist in the Hopf algebra \(A^*\), via dualization. We follow this recipe to write down explicit examples and show that it extends known techniques for defining twists. We also contribute with a detailed survey about twists in braided categories.


Hopf algebras Twists Braided twists 

Mathematics Subject Classification




We thank the referee for a careful reading of our article; Remarks 3.16 and 5.1 were introduced in response to his/her inquires and suggestions.


  1. 1.
    Aljadeff, E., Etingof, P., Gelaki, S., Nikshych, A.: On twisting of finite-dimensional Hopf algebras. J. Algebra 256(4), 484–501 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alonso Alvarez, J.N., Fernández Vilaboa, J.M.: Cleft extensions in braided categories. Commun. Algebra 28(7), 3185–3196 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andruskiewitsch, N.: On finite-dimensional Hopf algebras. In: Young Jang, S., Rock Kim Dae-Woong Lee, Y., Yie, I. (eds.) Proceedings of the International Congress of Mathematicians, Seoul 2014, vol. II, pp. 117–142 (2014)Google Scholar
  4. 4.
    Andruskiewitsch, N., Angiono, I., García Iglesias, A., Masuoka, A., Vay, C.: Lifting via cocycle deformation. J. Pure Appl. Algebra 218(4), 684–703 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Andruskiewitsch, N., Angiono, I., García Iglesias, A.: Liftings of Nichols algebras of diagonal type I. Cartan type A. Int. Math. Res. Not. IMRN. (in press). arXiv:1509.01622
  6. 6.
    Andruskiewitsch, N., Etingof, P., Gelaki, S.: Triangular Hopf algebras with the Chevalley property. Michigan Math. J. 49(2), 277–298 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Andruskiewitsch, N., Graña, M.: Examples of liftings of Nichols algebras over racks. AMA Algebra Montp. Announc. 2003(1). (2003)
  8. 8.
    Angiono, I., Kochetov, M., Mastnak, M.: On the rigidity of Nichols algebras. J. Pure Appl. Algebra 219, 5539–5559 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Angiono, I., Kochetov, M., Mastnak, M.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. 171(1), 375–417 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Andruskiewitsch, N., Vay, C.: Finite dimensional Hopf algebras over the dual group algebra of the symmetric group in three letters. Commun. Algebra 39, 4507–4517 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Angiono, I., García Iglesias, A.: Liftings of Nichols algebras of diagonal type II. All liftings are cocycle deformations. Submitted. arXiv:1605.03113
  12. 12.
    Ardizzoni, A., Beattie, M., Menini, C.: Cocycle deformations for Hopf algebras with a coalgebra projection. J. Algebra 324, 673–705 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ardizzoni, A., Beattie, M., Menini, C.: Cocycle deformations for liftings of quantum linear spaces. Commun. Algebra 39, 4518–4535 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ardizzoni, A., Beattie, M., Menini, C.: Braided bialgebras and quadratic bialgebras. Commun. Algebra 21, 1731–1749 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Doi, Y., Takeuchi, M.: Cleft comodule algebras for a bialgebra. Commun. Algebra 14, 801–817 (1986)MathSciNetMATHGoogle Scholar
  16. 16.
    Drinfeld, V.: Quantum Groups. Proc. of the ICM, Berkeley (1986)Google Scholar
  17. 17.
    Etingof, P., Gelaki, S.: On families of triangular Hopf algebras. Int. Math. Res. Not. IMRN 14, 757–768 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Etingof, P., Gelaki, S.: The classification of finite dimensional triangular Hopf algebras over an algebraically closed field of char 0. Mosc. Math. J. 3, 37–43 (2003)MathSciNetMATHGoogle Scholar
  19. 19.
    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor categories. Mathematical Surveys and Monographs 205 AMS, p. 344 (2015)Google Scholar
  20. 20.
    Fantino, F., García, G.A.: On pointed Hopf algebras over dihedral groups. Pacific J. Math. 252, 69–91 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Galindo, C., Natale, S.: Simple Hopf algebras and deformations of finite groups. Math. Res. Lett. 14(5–6), 943–954 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    García, G., Mastnak, M.: Deformation by cocycles of pointed Hopf algebras over non-abelian groups. Math. Res. Lett. 22, 59–92 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Garcí, G.A., Garcí, A.: Finite dimensional pointed Hopf algebras over \({\mathbb{S}}_4\). Israel J. Math. 183, 417–444 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    García Iglesias, A.: Representations of pointed Hopf algebras over \({\mathbb{S}}_3\). Rev. de la U. Mat. Arg. 51(1), 51–78 (2010)MathSciNetMATHGoogle Scholar
  25. 25.
    García Iglesias, A., Mombelli, M.: Representations of the category of modules over pointed Hopf algebras over \({\mathbb{S}}_3\) and \({\mathbb{S}}_4\). Pacific J. Math. 252(2), 343–378 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    García Iglesias, A., Vay, C.: Finite-dimensional pointed or copointed Hopf algebras over affine racks. J. Algebra 397, 379–406 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Grunelfelder, L., Mastnak, M.: Pointed and copointed Hopf algebras as cocycle deformations. (2007). arXiv:0709.0120
  28. 28.
    Klimyk, A.U., Schmüdgen, K.: Quantum Groups and Their Representations, Texts and Monographs in Physics. Springer, Berlin (1997)CrossRefMATHGoogle Scholar
  29. 29.
    Masuoka, A.: Extensions of Hopf algebras, Trabajos de Matemática 41/99. FaMAF (UNC) (1999)Google Scholar
  30. 30.
    Masuoka, A.: Defending the negated Kaplansky conjecture. Proc. Am. Math. Soc. 129, 3185–3192 (2001)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mombelli, M.: Families of finite-dimensional Hopf algebras with the Chevalley property. J. Pure Appl. Algebra 218, 2096–2118 (2014)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Montgomery, S.: Hopf Algebras and Their Action on Rings, CBMS Lecture Notes 82. American Math Society, Providence (1993)CrossRefGoogle Scholar
  33. 33.
    Movshev, M.: Twisting in group algebras of finite groups. Func. Anal. Appl. 27, 240–244 (1994)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Reshetikhin, N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20, 331–335 (1990)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Schauenburg, P.: Hopf bi-Galois extensions. Commun. Algebra 24, 3797–3825 (1996)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Sweedler, M.E.: Cohomology of algebras over Hopf algebras. Trans. Am. Math. Soc. 133, 205–239 (1968)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Takeuchi, M.: Free Hopf algebras generated by coalgebras. J. Math. Soc. Japan 22, 561–582 (1971)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2016

Authors and Affiliations

  • Nicolás Andruskiewitsch
    • 1
  • Agustín García Iglesias
    • 1
  1. 1.FaMAF-CIEM (CONICET), Universidad Nacional de CórdobaCórdobaArgentina

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